Mathematics Books

Book SynopsisThis book deals with the application of spectral methods to problems of uncertainty propagation and quanti?cation in model-based computations. It speci?cally focuses on computational and algorithmic features of these methods which are most useful in dealing with models based on partial differential equations, with special att- tion to models arising in simulations of ?uid ?ows. Implementations are illustrated through applications to elementary problems, as well as more elaborate examples selected from the authors’ interests in incompressible vortex-dominated ?ows and compressible ?ows at low Mach numbers. Spectral stochastic methods are probabilistic in nature, and are consequently rooted in the rich mathematical foundation associated with probability and measure spaces. Despite the authors’ fascination with this foundation, the discussion only - ludes to those theoretical aspects needed to set the stage for subsequent applications. The book is authored by practitioners, and is primarily intended for researchers or graduate students in computational mathematics, physics, or ?uid dynamics. The book assumes familiarity with elementary methods for the numerical solution of time-dependent, partial differential equations; prior experience with spectral me- ods is naturally helpful though not essential. Full appreciation of elaborate examples in computational ?uid dynamics (CFD) would require familiarity with key, and in some cases delicate, features of the associated numerical methods. Besides these shortcomings, our aim is to treat algorithmic and computational aspects of spectral stochastic methods with details suf?cient to address and reconstruct all but those highly elaborate examples.Table of ContentsIntroduction: Uncertainty Quantification and Propagation.- Basic Formulations.- Spectral Expansions.- Non-intrusive Methods.- Galerkin Methods.- Detailed Elementary Applications.- Application to Navier-Stokes Equations.- Advanced topics.- Solvers for Stochastic Galerkin Problems.- Wavelet and Multiresolution Analysis Schemes.- Adaptive Methods.- Epilogue.

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Book SynopsisAddressing students and researchers as well as Computational Fluid Dynamics practitioners, this book is the most comprehensive review of high-resolution schemes based on the principle of Flux-Corrected Transport (FCT). The foreword by J.P. Boris and historical note by D.L. Book describe the development of the classical FCT methodology for convection-dominated transport problems, while the design philosophy behind modern FCT schemes is explained by S.T. Zalesak. The subsequent chapters present various improvements and generalizations proposed over the past three decades. In this new edition, recent results are integrated into existing chapters in order to describe significant advances since the publication of the first edition. Also, 3 new chapters were added in order to cover the following topics: algebraic flux correction for finite elements, iterative and linearized FCT schemes, TVD-like flux limiters, acceleration of explicit and implicit solvers, mesh adaptation, failsafe limiting for systems of conservation laws, flux-corrected interpolation (remapping), positivity preservation in RANS turbulence models, and the use of FCT as an implicit subgrid scale model for large eddy simulations.Table of ContentsThe conception, gestation, birth and infancy of FCT.- The design of flux-corrected transport (FCT) algorithms for structured grids.- On monotonically integrated large eddy simulation of tubulent flows based on FCT algorithms.- Large scale urban simulations with FCT.- 30 years of FCT.- Algebraic flux corretion I.- Algebraic flux correction II.- Algebraic flux correction III.- Algebraic flux correction IV.- An evaluation of the FCT method for high-speed flows.- Flux-corrected and optimization-based remap.

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Book SynopsisOur book introduces a method to evaluate the accuracy of trend estimation algorithms under conditions similar to those encountered in real time series processing. This method is based on Monte Carlo experiments with artificial time series numerically generated by an original algorithm. The second part of the book contains several automatic algorithms for trend estimation and time series partitioning. The source codes of the computer programs implementing these original automatic algorithms are given in the appendix and will be freely available on the web. The book contains clear statement of the conditions and the approximations under which the algorithms work, as well as the proper interpretation of their results. We illustrate the functioning of the analyzed algorithms by processing time series from astrophysics, finance, biophysics, and paleoclimatology. The numerical experiment method extensively used in our book is already in common use in computational and statistical physics.Table of ContentsDiscrete stochastic processes and time series.- Trend definition.- Finite AR(1) stochastic process.- Monte Carlo experiments. - Monte Carlo statistical ensembles.- Numerical generation of trends.- Numerical generation of noisy time series.- Statistical hypothesis testing.- Testing the i.i.d. property.- Polynomial fitting.- Linear regression.- Polynomial fitting.- Polynomial fitting of artificial time series.- An astrophysical example.- Noise smoothing.- Moving average.- Repeated moving average (RMA).- Smoothing of artificial time series.- A financial example.- Automatic estimation of monotonic trends.- Average conditional displacement (ACD) algorithm.- Artificial time series with monotonic trends.- Automatic ACD algorithm.- Evaluation of the ACD algorithm.- A paleoclimatological example.- Statistical significance of the ACD trend.- Time series partitioning.- Partitioning of trends into monotonic segments.- Partitioning of noisy signals into monotonic segments.- Partitioning of a real time series.- Estimation of the ratio between the trend and noise.- Automatic estimation of arbitrary trends.- Automatic RMA (AutRMA).- Monotonic segments of the AutRMA trend.- Partitioning of a financial time series.

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Book SynopsisThis means that semigroup theory may be applied directly to the study of the equation I'!.f = h on M. In [45] Yau proves that, for h ~ 0, there are no nonconstant, nonnegative solutions f in [j' for 1 < p < 00. From this, Yau gets the geometric fact that complete noncom pact Riemannian manifolds with nonnegative Ricci curvature must have infinite volume, a result which was announced earlier by Calabi [4]. 6. Concluding Remarks In several of the above results, positivity of the semigroup plays an important role. This was also true, although only implicitly, for the early work of Hille and Yosida on the Fokker-Planck equation, i.e., Equation (4) with c = O. But it was Phillips [41], and Lumer and Phillips [37] who first called attention to the importance of dissipative and dispersive properties of the generator in the context of linear operators in a Banach space. The generation theorems in the Batty-Robinson paper appear to be the most definitive ones, so far, for this class of operators. The fundamental role played by the infinitesimal operator, also for the understanding of order properties, in the commutative as well as the noncommutative setting, are highlighted in a number of examples and applications in the different papers, and it is hoped that this publication will be of interest to researchers in a broad spectrum of the mathematical sub-divisions.Table of ContentsPositive Semigroups of Operators, and Applications: Editors’ Introduction.- Positive One-Parameter Semigroups on Ordered Banach Spaces.- Asymptotic Behavior of One-Parameter Semigroups of Positive Operators.- Positivity in Time Dependent Linear Transport Theory.- Quantum Dynamical Semigroups, Symmetry Groups, and Locality.- Stochastic Dilations of Uniformly Continuous Completely Positive Semigroups.- Order Properties of Attractive Spin Systems.- Book Reviews:.- E. B. Davies: One-Parameter Semigroups (William G. Faris).- L. Asimow and A. J. Ellis: Convexity Theory and its Applications in Functional Analysis (E. B. Davies).- Publications Received.- Announcement.

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Book SynopsisMathematics is often considered as a body of knowledge that is essen­ tially independent of linguistic formulations, in the sense that, once the content of this knowledge has been grasped, there remains only the problem of professional ability, that of clearly formulating and correctly proving it. However, the question is not so simple, and P. Weingartner's paper (Language and Coding-Dependency of Results in Logic and Mathe­ matics) deals with some results in logic and mathematics which reveal that certain notions are in general not invariant with respect to different choices of language and of coding processes. Five example are given: 1) The validity of axioms and rules of classical propositional logic depend on the interpretation of sentential variables; 2) The language­ dependency of verisimilitude; 3) The proof of the weak and strong anti­ inductivist theorems in Popper's theory of inductive support is not invariant with respect to limitative criteria put on classical logic; 4) The language-dependency of the concept of provability; 5) The language­ dependency of the existence of ungrounded and paradoxical sentences (in the sense of Kripke). The requirements of logical rigour and consistency are not the only criteria for the acceptance and appreciation of mathematical proposi­ tions and theories.Table of ContentsGeneral Philosophical Perspectives.- Logic, Mathematics, Ontology.- From Certainty to Fallibility in Mathematics?.- Moderate Mathematical Fictionism.- Language and Coding-Dependency of Results in Logic and Mathematics.- What is a Profound Result in Mathematics?.- The Hylemorphic Schema in Mathematics.- Foundational Approaches.- Categorical Foundations of the Protean Character of Mathematics.- Category Theory and Structuralism in Mathematics: Syntactical Considerations.- Reflection in Set Theory. The Bernays-Levy Axiom System.- Structuralism and the Concept of Set.- Aspects of Mathematical Experience.- Logicism Revisited in the Propositional Fragment of Le?niewski’s Ontology.- The Applicability of Mathematics.- The Relation of Mathematics to the Other Sciences.- Mathematics and Physics.- The Mathematical Overdetermination of Physics.- Gödel’s Incompleteness Theorem and Quantum Thermodynamic Limits.- Mathematical Models in Biology.- The Natural Numbers as a Universal Library.- Mathematical Symmetry Principles in the Scientific World View.- Historical Considerations.- Mathematics and Logics. Hungarian Traditions and the Philosophy of Non-Classical Logic.- Umfangslogik, Inhaltslogik, Theorematic Reasoning.

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Book SynopsisThe most important properties of normal and Student t-distributions are presented. A number of applications of these properties are demonstrated. New related results dealing with the distributions of the sum, product and ratio of the independent normal and Student distributions are presented. The materials will be useful to the advanced undergraduate and graduate students and practitioners in the various fields of science and engineering.Trade ReviewFrom the book reviews:“The aim of this book is to study the two most important continuous distributions, normal and Student’s t distribution. It is an interesting and well-written book with very clear exposition. The material presented is very useful both for students and researchers in various fields of science. … the book is very interesting and it will be useful for those who want to study normal and Student’s t distribution.” (Miroslav M. Ristić, Mathematical Reviews, February, 2015)Table of ContentsIntroduction.- Normal Distributions.- Student’s Distributions.- Sum, Product & Ratio for the Normal Random Variables.- Sum, Product & Ratio for the Student’s Random Variables.- Sum, Product & Ratio for the Normal and Student’s Random Variables.- Product of the Normal and Student’s Densities.- Characterizations of Normal Distributions.- Characterizations of Student’s Distributions.- Concluding Remarks.

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Book SynopsisMean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews. It is still in progress; fluid mechanics, gauge theory, plasma physics, quantum chemistry, mathematical oncology, non-equilibirum thermodynamics. spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature. The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics.Table of Contents

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Book SynopsisThe focus of this book is on providing students with insights into geometry that can help them understand deep learning from a unified perspective. Rather than describing deep learning as an implementation technique, as is usually the case in many existing deep learning books, here, deep learning is explained as an ultimate form of signal processing techniques that can be imagined. To support this claim, an overview of classical kernel machine learning approaches is presented, and their advantages and limitations are explained. Following a detailed explanation of the basic building blocks of deep neural networks from a biological and algorithmic point of view, the latest tools such as attention, normalization, Transformer, BERT, GPT-3, and others are described. Here, too, the focus is on the fact that in these heuristic approaches, there is an important, beautiful geometric structure behind the intuition that enables a systematic understanding. A unified geometric analysis to understand the working mechanism of deep learning from high-dimensional geometry is offered. Then, different forms of generative models like GAN, VAE, normalizing flows, optimal transport, and so on are described from a unified geometric perspective, showing that they actually come from statistical distance-minimization problems.Because this book contains up-to-date information from both a practical and theoretical point of view, it can be used as an advanced deep learning textbook in universities or as a reference source for researchers interested in acquiring the latest deep learning algorithms and their underlying principles. In addition, the book has been prepared for a codeshare course for both engineering and mathematics students, thus much of the content is interdisciplinary and will appeal to students from both disciplines.Trade Review“This book is based on material that has been prepared for senior-level undergraduate classes, this book can be used for one-semester senior-level undergraduate and graduate-level classes.” (Arzu Ahmadova, zbMATH 1493.68003, 2022)Table of ContentsPart I Basic Tools for Machine Learning: 1. Mathematical Preliminaries.- 2. Linear and Kernel Classifiers.- 3. Linear, Logistic, and Kernel Regression.- 4. Reproducing Kernel Hilbert Space, Representer Theorem.- Part II Building Blocks of Deep Learning: 5. Biological Neural Networks.- 6. Artificial Neural Networks and Backpropagation.- 7. Convolutional Neural Networks.- 8. Graph Neural Networks.- 9. Normalization and Attention.- Part III Advanced Topics in Deep Learning.- 10. Geometry of Deep Neural Networks.- 11. Deep Learning Optimization.- 12. Generalization Capability of Deep Learning.- 13. Generative Models and Unsupervised Learning.- Summary and Outlook.- Bibliography.- Index.

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Book SynopsisChapter 1  Distributions.- Chapter 2  Fundamental Solutions of Linear Differential Operators.- Chapter 3  Boundary Value Problems of the Laplace Equations.- Chapter 4  Boundary Value Problem of Modified Helmholtz Equation.- Chapter 5  Boundary Value Problems of Helmholtz Equation.- Chapter 6  Boundary Value Problems of the Navier Equations.- Chapter 7  Boundary Value Problems of the Stokes Equations.- Chapter 8  Some Nonlinear Problems.- Chapter 9  Coercive and Symmetrical Coupling Methods of Finite Element Method and Boundary Element Method.

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Book Synopsis1. Overview of Optimization Methods in VLSI Design.- 2. High-level Synthesis in VLSI Design.- 3. Optimization Methods in Physical Design.- 4. Power Minimization and Power Grid Synthesis.- 5. Efficient Testing and Verification Methods.- 6. Optimization Approaches for Clocking and Delay Minimization.- 7. VLSI Circuits and Approximate Computing.- 8. Challenges in Full Chip Optimization.

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Book SynopsisDielectric constant and fluctuation formulae for molecular dynamics.- PoissonBoltzmann electrostatics and analytical approximations.- Numerical methods for PoissonBoltzmann equations.- Random walk stochastic methods for boundary value problems.- Deep Neural Network for Solving PDEs.- Fast algorithms for long-range interactions.- Fast multipole methods for long-range interactions in layered media.- Maxwell equations, potentials, and physical/artificial boundary conditions.- Dyadic Green's functions in layered media.- High-order methods for surface electromagnetic integral equations.- High-order hierarchical Nedelec edge elements.- Time-domain methods discontinuous Galerkin method and Yee scheme.- Scattering in periodic structures and surface plasmons.- Schr odinger equations for waveguides and quantum dots.- Quantum electron transport in semiconductors.- Non-equilibrium Green's function (NEGF) methods for transport.- Numerical methods for Wigner quantum transport.- Hydrodynamic electron transport and finite difference methods.- Transport models in plasma media and numerical methods.

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Book SynopsisBackground in Potential Theory.- Fundamentals of Fine Potential Theory.- Further Developments.- Fine Complex Potential Theory.

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Book SynopsisThe Integration before Riemann.- The Definition of Integral Given by Riemann.- Geometric Definition of the Integral.- The Functions of the Bounded Variation.- The Search for Primitive Functions.- The Integration Defined with the Help of Primitive Functions.- The Definite Integral of Summable Functions.- The Indefinite Integral of Summable Functions.- The Seach of Primitive Functions. The Existence of Derivatives.- The Totalisation.- The Integral of Steiltjes.

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Book Synopsis6 Nodal Enriques Surfaces.- 7 Reye Congruences.- 8 Automorphisms of Enriques Surfaces.- 9 Rational Coble Surfaces.- 10 Supersingular K3 Surfaces and Enriques Surfaces.

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Book SynopsisChapter 1 Riemannian Geometry.- Chapter 2 Finslerian Geometry.- Chapter 3 Isometric immersions.

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Book Synopsis

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Book SynopsisChapter 1 Circular Data Diagnostics in Regression and Time Series Models.- Chapter 2 The Weighting Approach on the Circle.- Chapter 3 Isotonic Regression Estimators for Simultaneous Estimation of Order Restricted Location/Scale Parameters of a Bivariate Distribution: A Unified Study.

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Book SynopsisChapter 1.Preliminaries.- Chapter 2. Compact Complex Surfaces.- Chapter 3. Elliptic Surfaces and Lefschetz Fibrations.- Chapter 4. Non-Kähler Complex Structures on R2?.- Chapter 5.  Strongly Pseudoconvex Manifolds.- Chapter 6.  Contact Structures.- Chapter 7. Strongly Pseudoconcave Surfaces and Their Boundaries.

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Book SynopsisGlobally integrable quantum systems and their perturbations.- On two-dimensional Dirac operators with $delta$-shell interactions supported on unbounded curves with straight ends.- Attractor Subspace and Decoherence-Free Algebra of Quantum Dynamics.- Algebraic localization of generalized Wannier bases implies Roe triviality in any dimension.- Hearing the boundary conditions of the one-dimensional Dirac operator Bosonized Momentum Distribution of a Fermi Gas via Friedrichs Diagrams.- Self-adjointness and Domain of Generalized Spin–Boson Models with Mild Ultraviolet Divergences.- Random Linear Systems with Quadratic Constraints: from Random Matrix Theory to replicas and back.- New analytical and geometrical aspects on Trudinger-Moser type inequality in 2D.- Resolvent limits of exterior boundary value problems and singular perturbation of Laplace operator in 3D.- The Search for NLS Ground States on a hybrid domain: motivations, methods, and results.- From microscopic to macroscopic: the large number dynamics of agents and cells, possibly interacting with a chemical background.- Open problems and perspectives on solving Friedrichs systems by Krylov approximation.- Singularity: a Seventh Memo.

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Book SynopsisIntroduction.- Exponential Family of Distributions.- Univariate GLM.- Estimation and Tests for Univariate GLM.- Quasi Likelihood.- Multivariate Data and GLM: Generalized Estimating Equations.- Generalized Linear Mixed Models.- Extension of GLM for Bivariate Data.- Extension of GLM for Multivariate Data: Alternative Models.- Generalized Quasi Likelihood Methods.- Bayesian Approach for GLM.- GLM for Big Data Analytics.

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